2

$\mathsf{ZFC}$ is often introduced in logic textbooks as a first-order theory with equality and a single non-logical symbol $\in$. However, even stating the axioms of $\mathsf{ZFC}$ in this language is a cumbersome task: for instance, while the axiom of choice is often written as $$ \forall X \left( \varnothing \notin X \implies \exists f \colon X \rightarrow \bigcup X \quad \forall a \in X \, ( f(a) \in a ) \right) $$ this uses the symbols $\varnothing$, $\bigcup$, and function notation, and the "official" statement of the axiom of choice is a much longer string of symbols.

It seems that even statements which feel utterly trivial take many lines to be proven; for instance, in this post, there is a 25 line natural deduction proof that the empty set is unique. As far as I know, very few proofs in ordinary mathematics have actually been converted into formal proofs in $\mathsf{ZFC}$. For instance, I have never seen a formal proof of the consistency of $\mathsf{PA}$, even though it is universally agreed by logicians and set theorists that it is a consequence of the $\mathsf{ZFC}$ axioms.

So my question is: why are we confident that "ordinary proofs" can be converted into formal proofs in $\mathsf{ZFC}$ when so few actually are? In other words, why are we confident that the only thing stopping us from doing this conversion in practice is the sheer tediousness of it all?

Joe
  • 19,636
  • 7
    General model-theoretic results - see expansions by definitions - tell us that as far as logical strength goes, there's no change to adding "abbreviations" to ZFC. So the example about the axiom of union is a red herring, albeit of an instructive type. – Noah Schweber Jun 12 '23 at 02:31
  • 5
    Do you not trust a computer program because you've never seen it written out in 0's and 1's? – Lee Mosher Jun 12 '23 at 02:50
  • 6
    Why are we confident that the interactions of atoms ultimately make our buildings stand if very few buildings are actually constructed using nuclear physics? – Z. A. K. Jun 12 '23 at 05:29
  • 1
    "very few proofs in ordinary mathematics have actually been converted into formal proofs..." Yes, but no example of a math proof that cannot be formally rewritten has been found yet. – Mauro ALLEGRANZA Jun 12 '23 at 06:42
  • 2
    When I was studying the Axiom of Choice, I wanted to prove something that actually used the Axiom in its FOL form (No book or Internet source I had seen had attempted this). I proved "If $f$ is a surjection from $A$ to $B$ then there is an injection from $B$ to $A$", which I thought would be one of the simpler meaningful theorems which requires AC. The proof is 369 lines long! In addition, it's quite a bit of work creatingthe proof (a lot more than the wordy version!). – Porky Jun 12 '23 at 07:28

1 Answers1

3

As Noah Schweber points out, results like the conservativity of expansions-by-definitions give us some confidence. But I think there's more to it than that. On the other hand, as always with informal-to-formal matters (e.g., Church's Thesis), we cannot expect to prove that formalization is always possible.

First a couple of analogies. In topology, one often runs into claims about continuity based on geometric intuition (e.g., the "house with two rooms", Hatcher p.4). No one doubts that these could be reformulated as $\epsilon$-$\delta$ proofs, but nobody would want to carry out the rewriting. In computability theory, one can tell that certain functions are computable without actually writing programs, let alone constructing Turing machines.

In these and similar cases, the process of formalization is routine without being algorithmic. Once you've gained experience, you know what tools to deploy, and what pitfalls to worry about. To quote from an MO answer to the question of formalizing Wiles' proof in Peano Arithmetic:

Experts can "smell" the intrusion of a strong axiom from a long way off; for example, the primary reason that the Robertson-Seymour graph minor theorem requires more than PA is that it invokes a highly sophisticated argument by induction at a crucial point. The proof of FLT doesn't give off this sort of odor anywhere. Of course, a smell test is not a proof, but the smart money is that if we run into difficulties trying to prove FLT in PA, it won't be because we need some strong arithmetic axiom.

We should not forget that both the predicate calculus and the ZFC axioms arose from research programs of formalizing mathematical reasoning. For the predicate calculus, the Completeness Theorem gives us assurance that nothing was "left out". For ZFC, we have no comparable formal result, but the intuition behind the Cumulative Hierarchy is very suggestive.

Drilling down a bit, what makes this formalization process seem routine? I think it's because it is itself a "drilling down". Let's look at your example, the proof in ZFC of Con(PA). If one had to formalize it, one would start by breaking it into parts. At the top-level, the proof consists of two parts:

  1. The Soundness Theorem: if a theory $T$ has a model whose domain is a set, then it is consistent.
  2. PA has such a model in ZF, namely the finite ordinals.

What are key issues in formally proving the Soundness Theorem? They split into syntax and semantics. For syntax, all the pieces are "finite combinatorial structures": proofs are sequences of formulas, formulas sequences of symbols, etc. We have a great deal of confidence that ZFC can handle such things without even breathing hard.

Semantics seems prima facie like a bigger deal. But here again we have a roadmap: Tarski's inductive definition of truth (aka satisfaction), a piece of ordinary mathematics, not particulary convoluted. General results of ZFC handle the rather vanilla inductions and recursive definitions. I had occasion to write up a more detailed treatment; in it, I made free use of the ability of ZF to handle trees formally. If that were challenged, one would drill down into the details with resignation, perhaps, but without trepidation.

In Kaye's Models of Peano Arithmetic, he dealt with the more difficult problem of formalizing certain notions of truth inside PA. He starts with the sentence, "This is the chapter that no one wanted to have to write..." Nonetheless, working out the details at no point required real inspiration, just persistence. (The one non-obvious trick, representing sequences of arbitrary length in PA, was provided by Gödel long ago.)

Ultimately, drilling down should reach bedrock: something formalizable in ZFC. Of course you can't prove that it will, short of doing it. But it would be quite surprising if it didn't.

All that said, surprises can occur. For example, a theory I call ZFC$\neg\infty$ -- that's ZFC with the axiom of infinity replaced with its negation -- is, according to folklore, "equivalent" to PA. But when Kaye and Wong looked into the details, they found that for a strong notion of equivalence, ZFC$\neg\infty$ must be supplemented with another axiom. (See here for more details. The Kaye-Wong paper is "On Interpretations of Arithmetic and Set Theory", Notre Dame Journal of Formal Logic, 2007.)

More examples. Much of modern category theory uses Grothendieck Universes; these are usually justified by appealing to a large cardinal axiom. Finally, as I understand it, it is still not settled if Wiles' proof can be formalized in PA. The answer to this question remarks, "Wiles' original proof of FLT uses set-theoretical assumptions unprovable in Zermelo-Fraenkel set theory with axiom of choice", but goes on to quote a paper saying "certainly much less than ZFC is used in principle, probably nothing beyond PA, and perhaps much less than that."

Michael Weiss
  • 8,813

  • but nobody would want to carry out the rewriting. . It's worth pointing out that, if there was an argument where many mathematicians were claiming that a certain function was obviously continuous, but I had good reason to believe that their claim could not be reformulated as an ϵ-δ argument, I'd in fact be very eager to try to carry out that rewriting, show where it fails, and reap the rewards. But such situations are very rare indeed.

     – Z. A. K. Jun 12 '23 at 14:48
  • 2
    @Z.A.K. Quite true. But even in such a case, I’d expect you’d be able to pinpoint the fallacious assumption people were making without descending to the $\epsilon-\delta$ level. And to get your paper published, you’d need to do more than say, I tried to come up with an $\epsilon-\delta$ proof but couldn’t. – Michael Weiss Jun 12 '23 at 15:23
  • @MichaelWeiss: Thank you for this answer. Months after reading your answer, I stumbled on an argument for why "ordinary mathematical proofs" should be formalisable, in principle (this assertion is sometimes called Hilbert's Theisis). To me, at least, it seems quite different to the argument that you use, and so I wonder if you have any thoughts on it. – Joe Nov 07 '23 at 13:18
  • The argument (taken from Computability and Logic by Boolos, Burgess, and Jeffrey) is: "Suppose there is a proof in the ordinary mathematical sense of some theorem from some axioms. As part-time orthodox mathematicians ourselves, we presume ordinary mathematical methods of proof are sound, and if so, then the existence of an ordinary mathematical proof means that the theorem really is a consequence of the axioms... – Joe Nov 07 '23 at 13:20
  • ...But if the theorem is a consequence of the axioms, then the completeness theorem tells us that, in agreement with Hilbert’s thesis, there will be a formal deduction of the theorem from the axioms." – Joe Nov 07 '23 at 13:20
  • The weak point of their argument is knowing for certain that only the axioms of ZFC have been used in the “proof in the ordinary mathematical sense”. How can you be 100% sure without formalizing the proof in ZFC? Maybe some highly plausible assumption was used that you think (for example) is an instance of Replacement, but you find out when you try to formalize the argument that it isn’t. I regard that as the essence of the question, so the Boolos argument is actually addressing some other issue. – Michael Weiss Nov 07 '23 at 15:10
  • Specifically, their issue is the adequacy of the inference rules and the logical axioms of first-order predicate calculus. The adequacy of a specific set of axioms, namely ZFC,, to support all of ordinary mathematical reasoning, is an orthogonal question. – Michael Weiss Nov 07 '23 at 16:13
  • @MichaelWeiss: I see. Thank you for addressing the argument. One other doubt that I have is about the ordinary methods of proof being sound. When I write an informal proof about elementary set theory using the axioms of $\mathsf{ZFC}$, I usually informally have this idea that I am proving a fact about the "real" universe of sets, and the "real" membership relation; I don't ever stop to think whether the statement is true for foreign-seeming things like countable models of $\mathsf{ZFC}$ whose relation is not actually set membership. – Joe Nov 07 '23 at 17:20
  • Therefore, I don't feel fully certain that the statement I have proven is actually true for every model of $\mathsf{ZFC}$. I suspect that my doubts are misplaced, but I feel that I don't have a good enough handle on models of $\mathsf{ZFC}$ to be certain that ordinary methods of proof are sound in this case. – Joe Nov 07 '23 at 17:20
  • Apologies if it feels like I am pestering you with these comments. – Joe Nov 07 '23 at 17:26
  • Not at all. If your informal proof really can be formalized in ZFC, then it holds for all models of ZFC, by the very definition of “model of ZFC”, plus the soundness of the predicate calculus. As to when one can be confident that this formalization can be carried out without actually doing it, that brings us back to your original question and my answer. – Michael Weiss Nov 07 '23 at 17:49